At a high level, we translate the problem into an infinite LP, with the “variables” being the distribution \(\Sigma \)  over \(\Delta *{\Omega }\) . We first prove that the target function of the infinite LP is upper semi-continuous. Secondly, we show, using infinite-dimensional optimization tools, that it must attain a maximum at an extreme point of the feasible set. Thirdly, we argue that every extreme point has a finite support of bounded size, analyzing the effect of adding the Bayes-plausibility constraints one by one by considering the hyperplanes specifying the constraints: the maximal support size of extreme points is at most doubled upon each addition. Finally, we improve the bound on the support size of each extreme point using a finite LP.If the ex post constraints are convex, the result follows directly from the concavification approach of [1]}.
[Proof of Theorem REF ]
Denote the function and the constant specifying the \(i\) -th ex post constraint (\(1\le i\le m\) ) by \(i\)  and \(c_i\) , respectively. Let \(K:=\cap _{1\le i\le m} f_i^{-1}*{-\infty , c_i}\subseteq \Delta *{\Omega }\)  be the set of posteriors that are allowed to belong to a support of a valid signaling scheme. As \(f_1,...,f_m\)  are continuous, \(K\)  is compact.
